Compressed-sensing (CS)-based Image Deblurring Scheme …Uikyu Je, et al：CS-based Image Deblurring Scheme with a TV Regularization Penalty for Improving Image Characteristics in

Original Article PROGRESS in MEDICAL PHYSICS Vol 27 No 1 March 2016

httpdxdoiorg1014316pmp20162711

- 1 -

This work was supported by the Radiation Technology Development

Program of the National Research Foundation (NRF) funded by the

Korea Ministry of Science ICT amp Future Planning under Contract No

2015-51-0284

Received 10 March 2016 Revised 25 March 2016 Accepted 28

March 2016

Correspondence Hyosung Cho (hscho1yonseiackr)

Tel 82-33-761-9660 Fax 82-33-761-9664cc This is an Open-Access article distributed under the terms of the Creative Commons

Attribution Non-Commercial License (httpcreativecommonsorglicensesby-nc40) which

permits unrestricted non-commercial use distribution and reproduction in any medium

provided the original work is properly cited

Compressed-sensing (CS)-based Image Deblurring Scheme with a Total Variation Regularization Penalty for Improving

Image Characteristics in Digital Tomosynthesis (DTS)

Uikyu Je Kyuseok Kim Hyosung Cho Guna Kim Soyoung Park Hyunwoo Lim Chulkyu Park Yeonok Park

Department of Radiation Convergence Engineering and i TOMO Research Group Yonsei University Wonju Korea

In this work we considered a compressed-sensing (CS)-based image deblurring scheme with a total-variation

(TV) regularization penalty for improving image characteristics in digital tomosynthesis (DTS) We implemented

the proposed image deblurring algorithm and performed a systematic simulation to demonstrate its viability We

also performed an experiment by using a table-top setup which consists of an x-ray tube operated at 90 kVp

6 mAs and a CMOS-type flat-panel detector having a 198-μm pixel resolution In the both simulation and

experiment 51 projection images were taken with a tomographic angle range of θ=60o and an angle step of

Δθ=12o and then deblurred by using the proposed deblurring algorithm before performing the common

filtered-backprojection (FBP)-based DTS reconstruction According to our results the image sharpness of the

recovered x-ray images and the reconstructed DTS images were significantly improved and the cross-plane

spatial resolution in DTS was also improved by a factor of about 14 Thus the proposed deblurring scheme

appears to be effective for the blurring problems in both conventional radiography and DTS and is applicable

to improve the present image characteristics985103985103985103985103985103985103985103985103985103985103985103985103985103985103985103985103985103985103985103985103Key Words Compressed-sensing (CS) Digital tomosynthesis (DTS) Deblurring Total variation (TV)

Introduction

Digital tomosynthesis (DTS) has been popularly used as a

multiplanar imaging modality in medical and industrial x-ray

imaging applications It is a geometric tomography by a lim-

ited-angle scan to provide cross-sectional images of the scan-

ned object with a moderate cross-plane resolution12) However

although it provides some of the tomographic benefits of com-

puted tomography (CT) at a reduced imaging dose and time

the image characteristics is relatively inferior mainly due to

the blur artifacts originated from incomplete data sampling for

a limited angular range and also aspects inherent to imaging

system such as finite focal spot of the x-ray source detector

resolution system noise etc Thus the recovery of x-ray im-

ages from their blurred and noisy version collectively referred

to as image deblurring has been extensively studied during

the past decades

In this work we considered a compressed-sensing (CS)-

based deconvolution method for image deblurring of high ac-

curacy in DTS Here the CS is the state-of-the-art mathemat-

ical theory for solving the inverse problems which exploits

the sparsity of the image with substantially high accuracy3)

We implemented the proposed deblurring algorithm and per-

formed a systematic simulation and experiment to demonstrate

its viability for improving the image characteristics In the fol-

lowing sections we briefly describe the implementation of the

proposed deblurring algorithm the simulation and experimental

Uikyu Je et alCS-based Image Deblurring Scheme with a TV Regularization Penalty for Improving Image Characteristics in DTS

- 2 -

Fig 1 Schematic illustration of a typical DTS geometry in which an x-ray source and a flat-panel detector move together in an arc

around the pivot during the projection data acquisition

setup and we present the obtained results

Materials and Methods

1 CS-based image deblurring scheme

Most of the previous image deblurring methods are based

on the standard image degradation modeling in which the ob-

served image g(xy) is formed by convolving the original (or

exact) image f(xy) by the shift-invariant point-spread function

(PSF) of the system psf(xy) followed by adding white

Gaussian noise n(xy)

g(xy)=f(xy)otimesotimespsf(xy)＋n(xy) (1)

where the operator otimesotimes represents two-dimensional (2D)

convolution Thus the degraded image can be restored by the

deconvolution method provided that PSF is known4) The PSF

is the 2D image profile of a point object which describes the

amount of blurring by the imaging system Numerous decon-

volution methods such as Wiener filtering least-squares filter-

ing wavelet-based algorithms etc have already been estab-

lished in the literature56) However as indicated in (1) one

critical difficulty in the deconvolution problem is associated

with the presence of noise in the blurred image which results

in imperfect deconvolution

In this work as a highly accurate deconvolution method we

employed a CS-based image deblurring scheme incorporated

with the total-variation (TV) regularization penalty Here the

TV is the l1-norm of the gradient image and is often used as

an effective regularization penalty in many image recovery

problems such as denoising deblurring and inpainting due to

its ability to preserve image edges7) In the CS-based deblur-

ring scheme the original image f is recovered normally by

minimizing the following objective function ϕ (f) as the sol-

ution to a convex optimization problem f described in (2)

assuming that the negligible components of the gradient image

are nearly zeros

ϕ (f)=∥otimesotimes∥ ∥∥

f =arg min ϕ (f) (2) isin

where ∥otimesotimes∥ is the fidelity term ∥∥ is

the TV regularization penalty term α is the parameter that

balances the two terms and is chosen so that signal-to-noise

ratio is maximized (eg α=004 was used in this work) and

Q is the set of feasible f The convex optimization problem

described in (2) can be solved approximately but efficiently

by using the accelerated gradient-projection-Barzilai-Borwein

(GPBB) method8)

Fig 1 shows the schematic illustration of a typical DTS ge-

ometry in which an x-ray source and a flat-panel detector move

together in an arc around the pivot during the projection data

acquisition Fig 2 shows the simplified flowchart of the

CS-based deblurring scheme in DTS implemented in this study

Briefly to describe firstly the projection images of an imaged

PROGRESS in MEDICAL PHYSICS Vol 27 No 1 March 2016

- 3 -

Fig 2 Simplified flowchart of the

CS-based deblurring scheme in

DTS The acquired projection

images are deblurred through the

CS-based scheme before per-

forming the common FBP-based

DTS reconstruction

Fig 3 (a) 3D numerical phantom

of a pump casting used in the

simulation and (b) the table-top

setup that we established for the

experiment

object are acquired from the imaging system and then they are

assumed as the initial gauss f(0)

before applying the CS-based

framework as the current updated images f(k)

By using the f(k)

and the psf measured separately the objective function ϕ (f) is

computed to determine an optimal step size Δf(k)

for the next

updated images f(k+1)

The images f are successively updated

until the mismatch between the current and the next updated

images converges to a specified tolerance ε before performing

the common filtered-backprojection (FBP)-based DTS reconst-

ruction More detailed descriptions of the FBP-based DTS re-

construction can be found in our previous paper9)

2 Simulation and experimental setup

Based on the above descriptions we implemented an effec-

tive CS-based deblurring algorithm by using the MatlabTM

(ver 83) programming language and performed a systematic

simulation and experiment to investigate the image charac-

teristics Fig 3 shows (a) the 3D numerical phantom of a

pump casting used in the simulation and (b) the table-top set-

up that we established for the experiment The table-top setup

mainly consists of an x-ray tube whose operating conditions

are 90 kVp and 6 mA a CMOS-type flat-panel detector having

a 198-μm pixel resolution and a sample holder In the both

simulation and the experiment 51 projection images were tak-

en with a tomographic angle range of θ=60o and an angle

step of Δθ=12o and then deblurred by using the CS-based

deblurring algorithm before performing the common FBP-

based DTS reconstruction


- 4 -

Fig 4 (a) Measured 1D LSF curve with a Gaussian fit and (b) the resultant 2D blur kernel having a filter size of 15times15 The same

blur kernel was used in the both simulation and the experiment

Fig 5 (a) Some examples of the

blurred (left) and the deblurred

(right) projection images of the

pump casting and (b) their

enlarged images indicated by the

box A in (a) Only three projec-

tion images (ie for θ= minus30o 0

and ＋30o) out of the 51 are

indicated for simplicity)

Table 1 Geometrical and reconstruction parameters used in

the simulation and experiment

Parameter Dimension

Source-to-pivot distance (SPD) 4428 mm

Pivot-to-detector distance (PDD) 2886 mm

Tomographic angle range (θ) 60o

Angle step (Δθ) 12o

Pixel size 0198 mm

Voxel size 0198 mm

Test object (dimension) Pump casting (426times240times198)

toy car (532times1064times582)

Deblurring algorithm CS-based

DTS algorithm FBP-based

Fig 4 shows (a) the measured 1D line-spread function (LSF)

curve with a Gaussian fit (standard deviation σ=0358 mm)

and (b) the resultant 2D blur kernel having a filter size of

15times15 Here we used a slit camera (Pro-Project Corp 05-101)

having a 10-μm width for measuring the 1D LSF The same


The resultant DTS images were reconstructed with a voxel size

of 0198 mmtimes0198 mmtimes0198 mm and voxel dimensions of

426times240times198 for the simulation and 532times1064times582 for the

experiment Although the implemented algorithm has not yet

been accelerated by the graphics processing unit (GPU) the to-

tal time for image deblurring and DTS reconstruction was less

than 20 minutes on a normal workstation Detailed geometrical

and reconstruction parameters used in the simulation and experi-

ment are listed in Table 1

Results and Discussion

Fig 5 shows (a) some examples of the blurred (left) and the


- 5 -

Fig 7 The enlarged DTS images

indicated by the box A in Fig 6

for no blurring blurring and

deblurring cases The phantom

slice image (upper left) is also

indicated as the reference

Fig 6 Some examples of the

reconstructed DTS images of the

pump casting from the front side

(top) to the rear side (bottom) for

no blurring blurring and deblur-

ring cases Only 4 slices out of

the 426 are indicated for simpli-

city The phantom slice images

(the leftmost) are also indicated as

the reference

deblurred (right) projection images of the pump casting and

(b) their enlarged images indicated by the box A in (a) Only

three projection images (ie for θ=minus30o 0 and ＋30o) out

of the 51 are indicated for simplicity Note that the deblurred

projection images are quite visible compared to the blurred im-

ages which demonstrates the viability of the proposed scheme

for image deblurring in x-ray imaging Fig 6 shows some ex-

amples of the reconstructed DTS images of the pump casting

from the front side (top) to the rear side (bottom) for no blur-

ring blurring and deblurring cases Here only 4 slices out of

the 426 are indicated for simplicity The phantom slice images

(the leftmost) are also indicated as the reference Fig 7 shows

the enlarged DTS images indicated by the box A in Fig 6

Also note that how the structures of the pump casting in the


- 6 -

Fig 8 An example of the recon-

structed DTS images of the toy

car (ie for x=306th

slice) for (a)

blurring and (b) deblurring cases

Fig 9 The measured SSP curves along the xndashaxis from the

reconstructed DTS images of the small thin-square phantom

Here the intensity profile of the phantom in x is also indicated

as the reference

deblurred DTS image is quite visible demonstrating the via-

bility of the proposed scheme for image deblurring in DTS as

well Fig 8 shows an example of the reconstructed DTS im-

ages of the toy car (ie for x=306th slice) for (a) blurring and

(b) deblurring cases which is similar to the simulation result

One thing that should be addressed is that the cross-plane

resolution in DTS is usually inferior to that in CT due to its

limited-angle scan which produces relatively worse tomo-

graphic features The cross-plane resolution in tomography is

often described by the slice-sensitivity profile (SSP) Similar to

the in-plane resolution the SSP describes the system response

to a Dirac delta function δ(x) in x (ie along the depth di-

rection of the imaging volume) In many cases the SSP curve

itself is replaced by the full-width at half-maximum (FWHM)

In order to evaluate the SSP characteristic we performed the

same DTS reconstruction by using a small thin-square (1times5times5)

phantom which is located at the mid-plane (ie x=51st slice)

of the imaging volume Fig 9 shows the measured SSP curves

along the xndashaxis from the reconstructed DTS images of the

small thin-square phantom Here the intensity profile of the

phantom in x is also indicated as the reference The measured

FWHM value for the deblurred DTS images was about 123

mm about 14 times smaller than that for the blurred images

indicating that the proposed deblurring scheme is also effective

for improving the cross-plane resolution in DTS

Consequently according to our simulation and experimental

results the CS-based deblurring scheme seems to be effective

for improving the image characteristics in DTS as well as in

x-ray imaging


- 7 -

디지털 단층합성 X-선 영상의 화질개선을 위한 TV-압축센싱 기반 영상복원기법 연구

연세 학교 방사선융합공학 학원 i TOMO연구

제의규ㆍ김규석ㆍ조효성ㆍ김건아ㆍ박소 ㆍ임 우ㆍ박철규ㆍ박연옥

본 연구에서는 디지털 단층합성 엑스선 상의 화질특성을 개선하기 해 TV-압축센싱 기반 상복원 기법을 제안한다

제안된 상복원 기법의 유효성을 검증하기 해 우선 련 상복원 알고리즘을 구 하 으며 이를 이용하여 련 시

뮬 이션 실험을 함께 수행하 다 실험을 해 일반 x-선 (90 kVp 6 mAs) CMOS형 평 형 검출기(198 μm 픽셀크

기)로 구성된 실험장치를 구성하 으며 제한된 각도 60o도에서 2o 간격으로 총 51장의 투상 상을 획득하고 제안된 알

고리즘으로 상복원을 수행한 후 필터링 역투사법(FBP)을 사용하여 디지털 단층합성 상을 구 하 다 본 연구에서

수행된 결과에 의하면 제안된 상복원 기법은 일반 엑스선 상 디지털 단층합성 상의 흐린 상화질을 선명하게

개선하고 한 디지털 단층합성 상의 깊이 분해능을 향상시키는 이 이 있음을 확인함으로써 기존 디지털 단층합성

상의 화질을 크게 개선할 수 있을 것으로 망된다

심단어 압축센싱 디지털 단층합성법 상흐림 제거 총 변량

Conclusion

In this work as an effective image recovering method we

proposed a CS-based image deblurring scheme incorporated

with the TV regularization penalty for improving the present

image characteristics in DTS Our results indicate that the im-

age sharpness of the recovered x-ray images and the re-

constructed DTS images were significantly improved and the

cross-plane spatial resolution in DTS was also improved by a

factor of about 14 which demonstrates the viability of the pro-

posed scheme for image deblurring of high accuracy in DTS

However during the CS-based iterative procedure each iter-

ation loop requires one forward-projecting and one back-

ward-projecting operations taking a much longer reconstruction

time which is the primary computational bottleneck especially

in 3D reconstruction Thus the graphics processing unit (GPU)

should be implemented in the proposed algorithm to accelerate

the reconstruction processing in the future

References

1 J Dobbins and D Godfrey Digital x-ray tomosynthesis cur-

rent state of the art and clinical potential Phys Med Biol 48(19)

R65-106 (2003)

2 F Xu L Helfen T Baumbach and H Suhonen

Comparison of image quality in computed laminography and

tomography Opt Express 20(2) 794-806 (2012)

3 K Choi J Wang L Zhu T S Suh S Boyd and L

Xing Compressed sensing based cone-beam computed to-

mography reconstruction with a first-order method Med Phys

(37) 5113-5125 (2010)

4 H Xu T Huang X Lv and J Liu The Implementation of

LSMR in Image Deblurring Appl Math Inf 8(6) 3041-3048

(2014)

5 M Figueiredo and R Nowak An EM algorithm for wave-

let-based image restoration IEEE Trans Image Process 12

906-916 (2003)

6 X Li Fine-granularity and spatially-adaptive regularization for

projection-based image deblurring IEEE Trans Image Process

20 971-983 (2011)

7 S Babacan R Molina and A Katsaggelos Parameter

estimation in TV image restoration using variational distribution

approximation IEEE Trans Image Process 17(3) 326-339

(2008)

8 J Park B Y Song J S Kim et al Fast compressed

sensing-based CBCT reconstruction using Barzilai-Borwein for-

mulation for application to on-line IGRT Med Phys 39(3)

1207-1217 (2012)

9 J E Oh H S Cho D S Kim S I Choi and U K

Je Application of digital tomosynthesis (DTS) of optimal deblur-

ring filters for dental x-ray imaging J of the Korean Phys Soc

60(6) 1161-1166 (2012)


- 2 -

Fig 1 Schematic illustration of a typical DTS geometry in which an x-ray source and a flat-panel detector move together in an arc

around the pivot during the projection data acquisition

setup and we present the obtained results

Materials and Methods

1 CS-based image deblurring scheme

Most of the previous image deblurring methods are based

on the standard image degradation modeling in which the ob-

served image g(xy) is formed by convolving the original (or

exact) image f(xy) by the shift-invariant point-spread function

(PSF) of the system psf(xy) followed by adding white

Gaussian noise n(xy)

g(xy)=f(xy)otimesotimespsf(xy)＋n(xy) (1)

where the operator otimesotimes represents two-dimensional (2D)

convolution Thus the degraded image can be restored by the

deconvolution method provided that PSF is known4) The PSF

is the 2D image profile of a point object which describes the

amount of blurring by the imaging system Numerous decon-

volution methods such as Wiener filtering least-squares filter-

ing wavelet-based algorithms etc have already been estab-

lished in the literature56) However as indicated in (1) one

critical difficulty in the deconvolution problem is associated

with the presence of noise in the blurred image which results

in imperfect deconvolution

In this work as a highly accurate deconvolution method we

employed a CS-based image deblurring scheme incorporated

with the total-variation (TV) regularization penalty Here the

TV is the l1-norm of the gradient image and is often used as

an effective regularization penalty in many image recovery

problems such as denoising deblurring and inpainting due to

its ability to preserve image edges7) In the CS-based deblur-

ring scheme the original image f is recovered normally by

minimizing the following objective function ϕ (f) as the sol-

ution to a convex optimization problem f described in (2)

assuming that the negligible components of the gradient image

are nearly zeros

ϕ (f)=∥otimesotimes∥ ∥∥

f =arg min ϕ (f) (2) isin

where ∥otimesotimes∥ is the fidelity term ∥∥ is

the TV regularization penalty term α is the parameter that

balances the two terms and is chosen so that signal-to-noise

ratio is maximized (eg α=004 was used in this work) and

Q is the set of feasible f The convex optimization problem

described in (2) can be solved approximately but efficiently

by using the accelerated gradient-projection-Barzilai-Borwein

(GPBB) method8)

Fig 1 shows the schematic illustration of a typical DTS ge-

ometry in which an x-ray source and a flat-panel detector move

together in an arc around the pivot during the projection data

acquisition Fig 2 shows the simplified flowchart of the

CS-based deblurring scheme in DTS implemented in this study

Briefly to describe firstly the projection images of an imaged


- 3 -







DTS reconstruction





experiment





By using the f(k)



for the next

























- 4 -














Parameter Dimension





Pixel size 0198 mm

Voxel size 0198 mm























- 5 -
















the reference
















- 6 -



car (ie for x=306th

slice) for (a)





as the reference





























x-ray imaging


- 7 -













Conclusion

















References



R65-106 (2003)







(37) 5113-5125 (2010)



(2014)



906-916 (2003)



20 971-983 (2011)




(2008)




1207-1217 (2012)




60(6) 1161-1166 (2012)


- 3 -







DTS reconstruction





experiment





By using the f(k)



for the next

























- 4 -














Parameter Dimension





Pixel size 0198 mm

Voxel size 0198 mm























- 5 -
















the reference
















- 6 -



car (ie for x=306th

slice) for (a)





as the reference





























x-ray imaging


- 7 -













Conclusion

















References



R65-106 (2003)







(37) 5113-5125 (2010)



(2014)



906-916 (2003)



20 971-983 (2011)




(2008)




1207-1217 (2012)




60(6) 1161-1166 (2012)


- 4 -














Parameter Dimension





Pixel size 0198 mm

Voxel size 0198 mm























- 5 -
















the reference
















- 6 -



car (ie for x=306th

slice) for (a)





as the reference





























x-ray imaging


- 7 -













Conclusion

















References



R65-106 (2003)







(37) 5113-5125 (2010)



(2014)



906-916 (2003)



20 971-983 (2011)




(2008)




1207-1217 (2012)




60(6) 1161-1166 (2012)


- 5 -
















the reference
















- 6 -



car (ie for x=306th

slice) for (a)





as the reference





























x-ray imaging


- 7 -













Conclusion

















References



R65-106 (2003)







(37) 5113-5125 (2010)



(2014)



906-916 (2003)



20 971-983 (2011)




(2008)




1207-1217 (2012)




60(6) 1161-1166 (2012)


- 6 -



car (ie for x=306th

slice) for (a)





as the reference





























x-ray imaging


- 7 -













Conclusion

















References



R65-106 (2003)







(37) 5113-5125 (2010)



(2014)



906-916 (2003)



20 971-983 (2011)




(2008)




1207-1217 (2012)




60(6) 1161-1166 (2012)


- 7 -













Conclusion

















References



R65-106 (2003)







(37) 5113-5125 (2010)



(2014)



906-916 (2003)



20 971-983 (2011)




(2008)




1207-1217 (2012)




60(6) 1161-1166 (2012)

Documents

Compressed-sensing (CS)-based Image Deblurring Scheme …Uikyu Je, et al：CS-based Image Deblurring Scheme with a TV Regularization Penalty for Improving Image Characteristics in